# Stochastic six-vertex model in a half-quadrant and half-line open ASEP

## TL;DR

This paper establishes KPZ universality for the ASEP in a half-space with open boundary conditions, revealing Tracy-Widom GOE fluctuations at the origin, using novel half-space Macdonald processes.

## Contribution

Introduces half-space Macdonald processes and connects them to the half-space stochastic six-vertex model and ASEP, providing new tools for analyzing KPZ universality in half-space systems.

## Key findings

- Height function fluctuations follow Tracy-Widom GOE distribution.
- First KPZ asymptotics result for half-space systems outside free-fermionic models.
- Develops new probabilistic measures on Young diagrams for half-space models.

## Abstract

We consider the asymmetric simple exclusion process (ASEP) on the positive integers with an open boundary condition. We show that, when starting devoid of particles and for a certain boundary condition, the height function at the origin fluctuates asymptotically (in large time $\tau$) according to the Tracy-Widom GOE distribution on the $\tau^{1/3}$ scale. This is the first example of KPZ asymptotics for a half-space system outside the class of free-fermionic/determinantal/Pfaffian models. Our main tool in this analysis is a new class of probability measures on Young diagrams that we call half-space Macdonald processes, as well as two surprising relations. The first relates a special (Hall-Littlewood) case of these measures to the half-space stochastic six-vertex model (which further limits to ASEP) using a Yang-Baxter graphical argument. The second relates certain averages under these measures to their half-space (or Pfaffian) Schur process analogs via a refined Littlewood identity.